So far there has been a shortage of physical, biological, social, mathematical, or whatever sort of systems, of independent interest, which have been shown to follow a rule of evolution which would qualify them as cellular automata. There is a result from the theory of symbolic dynamics that cellular automata define the continuous mappings with respect to a particular topology, but it is not known whether this has led to the detailed examination of any particular automata.
Some examples have been proposed; Preston and Duff[58] describe image processing applications, Wolfram's reprint collection[73] includes applications to nonlinear equations describing chemical reactions. It has often been proposed that the discretization of partial differential equations will render them into cellular automata. However, one has to wonder whether this approach gives any new insight or produces useful methods of solution. That is, the utility of the typical theorems of cellular automata theory such as the evolution into cycles, or the existence of Garden of Eden configurations, has to be contrasted with the existence theorems, stability criteria, and so on of traditional differential equation theory.
There is no doubt that if an actual cellular automaton were built mimicing Laplace's equation, for example, that it would be enormously useful. Still, each cell would have to have enough states to represent a real number to a significant accuracy and would have to have a capacity for at least addition and shifting. The number of such cells would have to be large enough to interpolate a plane area to a reasonable degree of accuracy. So, the cost of the design of an appropriate integrated circuit, and subsequent volume of sales, would have to be balanced, on the one hand, against the cost and utility of designing some other comparably complex circuit, and on the other against the ease of performing the same simulation in a contemporary single or parallel computer.